Invariants
| Level: | $80$ | $\SL_2$-level: | $16$ | ||||
| Index: | $48$ | $\PSL_2$-index: | $24$ | ||||
| Genus: | $0 = 1 + \frac{ 24 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 6 }{2}$ | ||||||
| Cusps: | $6$ (of which $2$ are rational) | Cusp widths | $1^{2}\cdot2\cdot4\cdot8^{2}$ | Cusp orbits | $1^{2}\cdot2^{2}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 8I0 |
Level structure
| $\GL_2(\Z/80\Z)$-generators: | $\begin{bmatrix}3&46\\4&21\end{bmatrix}$, $\begin{bmatrix}4&77\\43&38\end{bmatrix}$, $\begin{bmatrix}13&4\\42&79\end{bmatrix}$, $\begin{bmatrix}37&74\\62&41\end{bmatrix}$, $\begin{bmatrix}39&76\\36&7\end{bmatrix}$ |
| Contains $-I$: | no $\quad$ (see 40.24.0.cb.2 for the level structure with $-I$) |
| Cyclic 80-isogeny field degree: | $12$ |
| Cyclic 80-torsion field degree: | $192$ |
| Full 80-torsion field degree: | $245760$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 105 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 24 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle -\frac{1}{5}\cdot\frac{(x+4y)^{24}(x^{8}-288x^{7}y+128x^{6}y^{2}+7424x^{5}y^{3}+30720x^{4}y^{4}+57344x^{3}y^{5}+43008x^{2}y^{6}+8192xy^{7}+4096y^{8})^{3}}{x^{2}(x+4y)^{28}(x^{2}-2xy-4y^{2})^{8}(3x^{2}-16xy-32y^{2})}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 16.24.0-8.n.1.8 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 80.24.0-8.n.1.4 | $80$ | $2$ | $2$ | $0$ | $?$ |
This modular curve is minimally covered by the modular curves in the database listed below.